Let $L(t)=f(x,t)$ be the <likelihood function>. Since the <prior distribution> is $N(0,I_p)$, the <posterior density> has the form
$$
\mu(t)=\frac1ZL(t)\phi(t),
\qquad
\phi(t)=(2\pi)^{-p/2}e^{-\lVert t\rVert^2/2},
$$
where $Z$ is the <Bayesian model evidence>[normalizing constant].

Choose $0<\delta\leq1/2$ and use the <Gaussian autoregressive proposal reversible with respect to a standard normal distribution>
$$
s=\sqrt{1-2\delta}\,t+\sqrt{2\delta}\,Z_0,
\qquad Z_0\sim N(0,I_p).
$$
Thus
$$
q(\mathord\cdot\mid t)
=N\!\left(\sqrt{1-2\delta}\,t,2\delta I_p\right),
$$
whose <covariance matrix> is the required $2\delta I_p$. Reversibility with respect to the <standard normal distribution> says
$$
\phi(t)q(s\mid t)=\phi(s)q(t\mid s),
$$
so
$$
\frac{q(t\mid s)}{q(s\mid t)}=\frac{\phi(t)}{\phi(s)}.
$$
The <Metropolis–Hastings acceptance probability> therefore reduces to
$$
\rho(t,s)
=\min\left\{
\frac{L(s)\phi(s)}{L(t)\phi(t)}
\frac{\phi(t)}{\phi(s)},1
\right\}
=\boxed{\min\left\{\frac{f(x,s)}{f(x,t)},1\right\}}.
$$
This is the <Preconditioned Crank–Nicolson algorithm> with proposal scale $\beta=\sqrt{2\delta}$.

Solution

Past exam of the mathematics course of the University of Cambridge / 2020 / ii / Paper 3 / 28J / ii / Solution

{c}

= pCN algorithm
{c}
{synonym}

The preconditioned Crank–Nicolson algorithm, or pCN algorithm, uses a <Gaussian autoregressive proposal reversible with respect to a standard normal distribution> in a <Metropolis–Hastings algorithm>. When the target density is a likelihood times the proposal's invariant Gaussian prior density, the Gaussian factors cancel from the <Metropolis–Hastings acceptance probability>, leaving the likelihood ratio.

Preconditioned Crank–Nicolson algorithm

Let $0<\beta\leq1$, let $X\in\mathbb R^p$, and propose
$$
Y=\sqrt{1-\beta^2}\,X+\beta Z,
\qquad Z\sim N(0,I_p).
$$
The <proposal distribution> is $N(\sqrt{1-\beta^2}\,X,\beta^2I_p)$. If $X\sim N(0,I_p)$ independently of $Z$, then $(X,Y)$ is a jointly <multivariate normal distribution> invariant under exchanging $X$ and $Y$, because both <random vectors> have <covariance matrix> $I_p$ and their cross-covariance matrices are both $\sqrt{1-\beta^2}I_p$. Consequently its density $q$ satisfies
$$
\phi(x)q(y\mid x)=\phi(y)q(x\mid y),
$$
where $\phi$ is the standard-normal density. Thus the proposal is <reversible Markov chain>[reversible] with respect to the standard normal distribution.

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 Incoming links: Gaussian autoregressive proposal reversible with respect to a standard normal distribution